LEAST-SQUARES FINITE ELEMENT SOLUTION OF EULER EQUATIONS WITH ADAPTIVE MESH REFINEMENT

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

HAYRĐ YĐĞĐT AKARGÜN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

MECHANICAL ENGINEERING

FEBRUARY 2012

Approval of the thesis

LEAST-SQUARES FINITE ELEMENT SOLUTION OF EULER EQUATIONS WITH ADAPTIVE MESH REFINEMENT

Submitted by HAYRĐ YĐĞĐT AKARGÜN in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan ÖZGEN _______________ Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Suha ORAL _______________ Head of Department, Mechanical Engineering Assist. Prof. Dr. Cüneyt SERT _______________ Supervisor, Mechanical Engineering Dept., METU Examining Committee Members: Prof. Dr. Mehmet Haluk AKSEL _______________ Mechanical Engineering Dept., METU Assist. Prof. Dr. Cüneyt SERT _______________ Mechanical Engineering Dept., METU Inst. Dr. Tahsin ÇETĐNKAYA _______________ Mechanical Engineering Dept., METU Assist. Prof. Dr. Mehmet Metin YAVUZ _______________ Mechanical Engineering Dept., METU Ali AKGÜL, M.Sc. _______________ Chief Engineer, ROKETSAN

Date: _______________

iii

I hereby declare that all the information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last name :

Signature :

iv

ABSTRACT

LEAST-SQUARES FINITE ELEMENT SOLUTION OF EULER

EQUATIONS WITH ADAPTIVE MESH REFINEMENT

Akargün, Hayri Yiğit M.Sc., Department of Mechanical Engineering Supervisor : Assist. Prof. Dr. Cüneyt SERT

February 2012, 63 Pages

Least-squares finite element method (LSFEM) is employed to simulate 2-D and

axisymmetric flows governed by the compressible Euler equations. Least-squares

formulation brings many advantages over classical Galerkin finite element methods.

For non-self-adjoint systems, LSFEM result in symmetric positive-definite matrices

which can be solved efficiently by iterative methods. Additionally, with a unified

formulation it can work in all flight regimes from subsonic to supersonic. Another

advantage is that, the method does not require artificial viscosity since it is naturally

diffusive which also appears as a difficulty for sharply resolving high gradients in the

flow field such as shock waves. This problem is dealt by employing adaptive mesh

refinement (AMR) on triangular meshes. LSFEM with AMR technique is

numerically tested with various flow problems and good agreement with the

available data in literature is seen.

Keywords: Least-Squares Finite Element Method, Computational Fluid Dynamics,

Compressible Flow, Inviscid Flow, Euler Equations, Adaptive Mesh Refinement.

v

ÖZ

EN KÜÇÜK KARELER SONLU ELEMANLAR METODU ĐLE EULER

DENKLEMLERĐNĐN UYARLANMIŞ AĞLARDA ÇÖZÜMÜ

Akargün, Hayri Yiğit Yüksek Lisans, Makina Mühendisliği Bölümü Tez Yöneticisi : Yrd. Doç. Dr. Cüneyt Sert

Şubat 2012, 63 Sayfa

Sıkıştırılabilir Euler denklemlerinin modellediği iki boyutlu düzlemsel ve eksenel

simetrik akışların en küçük kareler sonlu elemanlar metodu kullanılarak benzetimi

yapılmıştır. En küçük kareler formülasyonunun geleneksel Galerkin sonlu elemanlar

metoduna göre birçok avantajı vardır. En küçük kareler formülasyonu kendine eş

olmayan denklemler için simetrik kesin artı matris oluşturur ve böylece sonuç sistemi

döngülü yöntemler ile verimli bir şekilde çözülebilir. Ayrıca, ses-altı hızlardan ses-

üstü hızlara kadar bütün akış rejimleri tek bir formülasyon ile çözülebilmektedir. Bu

metod kendiliğinden dağılımlı olduğu için yapay viskozite kullanımına gerek

duyulmamaktadır. Ancak metodun bu özelliği aynı zamanda akıştaki keskin

değişimlerin yüksek hassasiyet ile çözümlenmesini zorlaştırmaktadır. Bu sebepten

dolayı, akış çözücüsüne üçgen ağlar için çözüm ağı uyarlama algoritması

eklenmiştir. Geliştirilen çözücü farklı akış problemleri için sayısal olarak test

edilmiştir. Elde edilen sonuçlar literatürde ulaşılan sonuçlar ile uyumludur.

Anahtar Kelimeler: En Küçük Kareler Sonlu Elemanlar Metodu, Hesaplamalı

Akışkanlar Dinamiği, Sıkıştırılabilir Akışlar, Ağdasız Akışlar, Euler Denklemleri,

Çözüm Ağı Uyarlaması

vi

To My Family

vii

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest gratitude to my supervisor Dr.

Cüneyt Sert for his endless contributions and guidance throughout my thesis. In

every stage of this work, he has supported me with patience and been very friendly.

I also appreciate the suggestions, criticism and interpretations done on my work by

the thesis examining committee members.

I would especially want to thank my wife, Müge, for her understanding and support

throughout my study. I also want to express my gratitude to my parents for their

support.

I would also like to thank my chief engineer Ali Akgül for his advices and criticism

which have been very valuable to me. Additionally, I want to thank my colleagues in

Aerodynamics Division of Roketsan for their help and encouragement.

The last but not the least, I want to thank my friends and colleagues Mehmet Akgül

and Mustafa Uğur Göktolga for their support and friendship during this study.

viii

TABLE OF CONTENTS

ABSTRACT ............................................................................................................... iv

ÖZ ................................................................................................................................ v

ACKNOWLEDGEMENTS ..................................................................................... vii

TABLE OF CONTENTS ........................................................................................ viii

LIST OF FIGURES .................................................................................................. ix

INTRODUCTION ...................................................................................................... 1

1.1 Finite Element Method (FEM) ...................................................................... 2

1.2 Least-Squares Finite Element Method (LSFEM) .......................................... 4

1.3 Adaptive Mesh Refinement (AMR) .............................................................. 6

1.4 Current Study ................................................................................................ 9

GOVERNING EQUATIONS & LEAST SQUARES FEM FORMULATION . 10

2.1 Governing Equations ................................................................................... 10

2.2 Least-Squares Finite Element Formulation ................................................. 16

2.3 Lagrange Shape Functions .......................................................................... 18

2.4 Boundary Conditions ................................................................................... 20

2.4.1 Penalty Method ........................................................................................ 20

2.4.2 Coordinate Rotation Method ................................................................... 23

2.5 Adaptive Mesh Refinement (AMR) ............................................................ 24

2.5.1 Error Indicator.......................................................................................... 24

2.5.2 Element Division ..................................................................................... 24

RESULTS ................................................................................................................. 26

3.1 Supersonic Flow Over a Circular Bump ..................................................... 26

3.2 Transonic Flow Over a Circular Bump ....................................................... 36

3.3 Supersonic Ramp Flow ............................................................................... 44

3.4 Flow Over an Axisymmetric Body ............................................................. 52

3.5 Flow Through Scramjet Inlet ...................................................................... 55

CONCLUSION & FUTURE WORK ..................................................................... 58

4.1 Conclusion ................................................................................................... 58

4.2 Future Work ................................................................................................ 59

REFERENCES ......................................................................................................... 61

ix

LIST OF FIGURES

FIGURES

Figure 1-1: Linear triangular and quadrilateral elements ............................................. 2

Figure 1-2: Initial and refined (using r-refinement) meshes for supersonic bump flow

[24] ............................................................................................................................... 7

Figure 1-3: p-refinement of a quadrilateral element with hanging nodes shown in

black ............................................................................................................................. 7

Figure 1-4: h-refinement of a quadrilateral element with hanging nodes shown in

black ............................................................................................................................. 8

Figure 1-5: h-refinement of a triangular element without hanging nodes ................... 8

Figure 1-6: Demonstration of undesired mesh propagation problem of the longest

edge refinement algorithm [30] .................................................................................... 9

Figure 2-1: 4-node linear and 9-node quadratic quadrilateral master elements ......... 18

Figure 2-2: 3-node linear and 6-node quadratic triangular master elements ............. 19

Figure 2-3: Velocity components at a solid wall ....................................................... 21

Figure 2-4: Wall boundary node A ............................................................................ 21

Figure 2-5: Element division ...................................................................................... 25

Figure 3-1: Sketch of the supersonic flow over a circular bump problem ................. 27

Figure 3-2: Quadrilateral and triangular computational meshes used for supersonic

bump problem (2511 nodes) ...................................................................................... 28

Figure 3-3: Pressure contours for supersonic bump problem with ������ = 1.65 and � = 0.1: (a) linear quadrilateral elements (b) quadratic quadrilateral elements

(c) linear triangular elements (d) quadratic triangular elements ................................ 29

Figure 3-4: Final adapted mesh for supersonic bump problem.................................. 30

Figure 3-5: Zoomed view of the adapted mesh around the leading edge shock of

supersonic bump problem .......................................................................................... 30

x

Figure 3-6: Pressure contours of the final adapted solution of the supersonic bump

problem ...................................................................................................................... 31

Figure 3-7: Pressure data extraction from y=0.5 for the supersonic bump problem . 31

Figure 3-8: Comparison of pressure distribution at y=0.5 for 5 different solutions of

the supersonic bump problem .................................................................................... 32

Figure 3-9: Mach number distribution at the lower wall of the supersonic bump

problem ...................................................................................................................... 33

Figure 3-10: Mach number distribution at the upper (top) and lower wall (bottom) for

the supersonic bump problem .................................................................................... 34

Figure 3-11: Convergence histories of different solutions for the supersonic bump

problem ...................................................................................................................... 35

Figure 3-12: Sketch of the transonic flow over a circular bump problem ................. 36

Figure 3-13: Computational meshes of (top) rectangular and (bottom) triangular

elements, both having 2511 nodes ............................................................................. 37

Figure 3-14: Mach number contours for transonic bump problem with ������ = 0.675 (a) linear quadrilateral elements (b) quadratic quadrilateral elements

(c) linear triangular elements (d) quadratic triangular elements ................................ 38

Figure 3-15: Mach number distribution at the lower wall for different element types

and orders ................................................................................................................... 39

Figure 3-16: Initial coarse mesh for AMR ................................................................. 40

Figure 3-17: Initial (black) and adapted (red) mesh for the transonic bump problem40

Figure 3-18: Initial and adapted meshes and pressure contours near bump, � = 0.2:

(a) Initial mesh, (b) 2nd adaptation, (c) 4th adaptation, (d) 8th adaptation ................... 41

Figure 3-19: Mach number distribution at the lower wall of adapted solution of the

transonic bump problem ............................................................................................. 42

Figure 3-20: Mach number distribution at the exit of the transonic bump problem .. 43

Figure 3-21: Convergence histories of different solutions for the supersonic bump

problem ...................................................................................................................... 43

Figure 3-22: Sketch of the supersonic ramp flow ...................................................... 44

Figure 3-23: Uniform mesh for the supersonic ramp flow (7512 elements, 3872

nodes) ......................................................................................................................... 45

xi

Figure 3-24: Pressure contours of supersonic ramp flow obtained with different � 46

Figure 3-25: Effect of time step on pressure distribution at x=2.8 for supersonic ramp

flow ............................................................................................................................ 46

Figure 3-26: Coarse initial mesh for solving the supersonic ramp problem with AMR

(113 nodes and 194 elements) .................................................................................... 47

Figure 3-27: The grids and pressure contours for initial and adapted solutions of the

supersonic ramp problem: (a) Initial mesh, (b) 3rd adaptation, (c) 6th adaptation, (d)

9th adaptation .............................................................................................................. 48

Figure 3-28: Initial and final adapted mesh used for the supersonic ramp problem .. 49

Figure 3-29: Zoomed view of the initial and adapted mesh at shock location .......... 49

Figure 3-30: Pressure distribution for adapted and exact solution at � = 2.8 for the

supersonic ramp problem ........................................................................................... 50

Figure 3-31: Convergence history of the adapted solution of the supersonic ramp

problem ...................................................................................................................... 51

Figure 3-32: Axisymmetric body geometry (dimensions are in body diameters) ..... 52

Figure 3-33: Initial mesh around the axisymmetric body with 4092 nodes ............... 52

Figure 3-34: Adapted mesh and pressure contours of the axisymmetric flow problem

.................................................................................................................................... 53

Figure 3-35: Pressure distribution on the axisymmetric body ................................... 54

Figure 3-36: Sketch of the inlet problem (all dimensions are in meters) ................... 55

Figure 3-37: Initial mesh for the scramjet inlet problem with 1422 elements and 775

nodes .......................................................................................................................... 55

Figure 3-38: Adapted mesh and pressure contours of the scramjet inlet problem ..... 56

Figure 3-39: Pressure coefficient along the wall of the scramjet inlet ....................... 57

Figure 3-40: Convergence history of the adapted solution for the scramjet inlet

problem ...................................................................................................................... 57

1

CHAPTER 1

INTRODUCTION

In the past few decades, the progress in Computational Fluid Dynamics (CFD) has

been enormous. Due to increasing computational power and development of various

robust solution techniques, nowadays, many complex fluid dynamics problems can

be numerically simulated with acceptable accuracy. But there is always need for

further development in order to be able to solve large 3-D problems with limited

computational resources

In CFD, the most extensive equation of motion that describes a wide range of flow

conditions is the Navier-Stokes (N-S) equations. Considering today's computational

technology, direct numerical simulation of these equations is computationally

expensive and not practical for many real life problems of engineering interest. For

this reason, to simulate flow fields efficiently and accurately, some assumptions need

to be done, which depend on the problem to be solved. For example, when the

viscous forces are small compared to the inertial forces, such as high Reynolds

number flows, inviscid flow assumption is usually used. For such flows, away from

the boundary layer, compressible Euler equations can simulate the flow field in

essentially the same manner as the Navier-Stokes equations. In the current study 2-D,

compressible flow fields modeled by Euler equations are simulated.

Governing equations of flow problems can be solved numerically with different

methods such as finite volume method, finite difference method finite element

2

method, spectral method and boundary element method. In the current study finite

element method is used and its basic principles are explained in the next section.

1.1 Finite Element Method (FEM)

FEM is a numerical technique used to solve partial differential, integral and

variational equations. The first step in the FEM is discretizing the continuum into a

finite number of simple-shaped elements. The most commonly used elements in 2-D

are triangular and quadrilateral, shown in Figure 1-1. Polynomial order of shape

functions that are used to describe the variation of unknowns over these elements is

usually taken to be linear or quadratic in practice, but in theory it can be any desired

order. In FEM, governing equations of the problem are written over each element of

the solution domain one by one and individual elemental systems written in terms of

elemental nodal unknowns are obtained by variational or weighted residual

approaches. After obtaining elemental systems, they are assembled using element

connectivity information to form a global system of linear algebraic equations.

Finally, the global system is solved using an appropriate linear system solver.

Figure 1-1: Linear triangular and quadrilateral elements

The first use of FEM, in engineering sense, goes back to 1941 when Hrennikoff

developed "framework method" by dividing an elastic continuum structure into beam

elements [1]. In 1960, the term "finite element method" is first used by Clough in an

article [2]. Zienkiewicz published the first book about FEM in 1967 [3]. Since then,

3

FEM has started to be used in a wide variety of engineering fields including fluid

dynamics.

In FEM literature, many studies have been done for the solution of compressible

Euler equations. Many different versions of FEM and mesh refinement techniques

have been developed throughout the years. Hughes and Tezduyar presented Petrov-

Galerkin finite element formulation for the solution of one and two dimensional

compressible flows involving shocks [4]. Oden et al. used Taylor-Galerkin FEM for

solving high speed compressible flows with an adaptive mesh algorithm [5]. In their

study, artificial dissipation and flux-corrected transport scheme (FCT) is employed in

order to avoid oscillations in the flow field. Loth, Baum and Löhner also used this

method with FCT and artificial viscosity to solve high-speed axisymmetric nozzle

flows [6].

A popular version of FEM employed by Beau and Tezduyar is based on Streamline-

Upwind/Petrov-Galerkin (SUPG) formulation, which modifies the classical Galerkin

formulation by adding numerical stability [7]. Based on this method, Tezduyar et al.

introduced the deformable-spatial-domain/stabilized-space-time formulation which

involves solution of moving boundary problems [8]. Peraire et al. modified SUPG

formulation for solving compressible Euler equations at very low Mach numbers

where standard SUPG algorithm fails to provide accurate results [9]. Peraire et al.

also presented a higher order accurate FEM solution based on SUPG formulation for

compressible Euler and N-S equations and compared the numerical solutions

obtained by using linear and quadratic elements [10].

Bassi and Rebay used a higher-order discontinuous Galerkin FEM (DGFEM) to

solve 2-D Euler equations [11]. This method combines the features of finite element

and finite volume methods to develop a robust and accurate solver for flow problems

involving shocks. Hartmann and Houston used the same method with an adaptive

mesh refinement algorithm for compressible Euler equations and tested it with

transonic nozzle flow and supersonic flow over airfoil problems [12]. Tu and

4

Aliabadi used DGFEM with a slope limiting procedure in order to remove

oscillations at shock regions [13]. Authors claimed that added slope limiting

procedure results in a robust DG solver for compressible Euler equations. Another

DGFEM study presented by Feistauer and Kucera aims to solve compressible Euler

equations in a wide range of Mach numbers [14]. Solution of compressible flows

with DGFEM based methods is still an active research area [15-18].

1.2 Least-Squares Finite Element Method (LSFEM)

Another version of FEM, which is also used in the current study, is LSFEM. The

basic principle of this method is to minimize the residuals of a first-order differential

equation system in a least-squares sense [19]. As stated by Jiang the theoretical basis

of the least-squares method is the bounded inverse theorem of linear operators. Based

on this theorem, Jiang proved that LSFEM always leads to a convergent solution for

a well-posed first order partial differential equation system without need to apply any

special treatments such as upwinding or artificial dissipation.

There are significant advantages of least-squares finite element formulation

compared to the classical Galerkin finite element formulation [19]. Maybe the most

important one is the efficiency of the method. In LSFEM, non-self-adjoint equation

systems, one of which is the governing equations of fluid dynamics, result in

symmetric positive-definite global system matrices which can be solved efficiently

by iterative linear solvers with relatively less memory requirement and CPU time

compared to Galerkin FEM solutions [19]. This makes LSFEM especially more

suitable for large-scale problems without need of high computational resources.

Additionally, the theoretical basis of the least-squares minimization makes LSFEM a

robust technique that does not need any special treatments and does not have

problem dependent parameters to tune [20]. It can work in all flow regimes;

subsonic, transonic, supersonic and hypersonic with a unified formulation. This

advantage also extends to the field of structural mechanics, making LSFEM suitable

5

for fluid structure interaction problems. Recent increase of interest in LSFEM also

confirms these advantages. In the next section studies found in the literature to solve

compressible Euler equations with LSFEM are summarized.

For the solution of the compressible Euler equations Fletcher used the least-squares

method in primitive variables form [21]. Jiang and Carey used this method for the

solution of incompressible flows with adaptive mesh refinement (AMR) in which

least-squares functional is used as an error-indicator for AMR [22]. The resulting

system is solved by the element-by-element conjugate gradient method. In another

study, same authors extend their work for solving compressible Euler equations

without mesh adaptation [23]. They used two different schemes; one minimizes the �� norm of the residual and the other minimizes the weighted �� norm of the

residual with the backward-Euler time-discretization. In the �� norm based scheme,

numerical dissipation is further increased at the discontinuities by introducing an

additional term to the least-squares functional which is controlled by a user tuned

parameter to overcome the oscillations near the shock regions. Both methods are

tested with various problems and it is seen that �� scheme is a robust method but has

relatively poor shock resolution, whereas �� method has better shock resolution but

it needs user defined shock control parameter. Due to the numerical dissipation

inherent to LSFEM, shock waves cannot be resolved sharply on coarse grids. To

obtain high shock resolution, Habashi et al. used an r-type grid adaptation algorithm

[24]. Supersonic and transonic shock reflection and flow over a bump problems are

solved and flow features are shown to be captured accurately. Guaily and Megahed

also used LSFEM with r-type grid adaptation for 2-D planar and axisymmetric

compressible flows [25]. In addition to flow over a bump and shock-reflection

problems, they also solved nozzle flow with shocks and axisymmetric jet flow.

Moussaoui used a unified formulation based on LSFEM to calculate both very low

Mach number flows and compressible flows [26]. Moussaoui's approach is basically

to apply a new scaling technique for the flow variables which makes it numerically

appropriate for very low Mach numbers. Reddy et al. presented least-squares finite

element models for both high-speed inviscid and low-speed viscous compressible

6

flows [27]. For the solution of the compressible Euler equations, Gerritsma et al.

employed least-squares formulation with higher order spectral methods [28].

1.3 Adaptive Mesh Refinement (AMR)

In computational fluid dynamics, sometimes the initially created mesh can not

accurately resolve the flow features such as variations and discontinuities of the

variables especially for flows involving shock and expansion waves. In order to

simulate the variations in a flow field accurately, the initial mesh can be

adapted/modified until the desired accuracy is achieved. There are mainly three

methods used for grid adaptation which are r-refinement, h-refinement and p-

refinement. These methods can also be combined for hybrid adaptation purposes.

In r-refinement, mesh is modified by moving nodes without changing grid

connectivity and number of nodes/elements. Grid movement algorithm is usually

based on the spring analogy which basically considers each element edge as a spring

with a spring constant being a function of a user selected error indicator. Figure 1-2

taken from Habeshi et. al [24] demonstrates r-refinement for supersonic flow over a

bump problem. As seen from Figure 1-2, grids are moved to the shock regions to

accurately capture discontinuities in the flow field. But this method is highly

dependent on the initial mesh because the extend of refinement is limited to the

available nodes in the domain.

In p-refinement, higher order polynomial interpolations are used on the elements to

be refined without changing the number of elements. Even though the number of

elements is not changed, resultant mesh has non-conformal elements with hanging

nodes. An example to p-refinement of a 2-D quadrilateral element is given in Figure

1-3, in which the order of upper left element is increased from 1 to 2.

7

Figure 1-2: Initial and refined (using r-refinement) meshes for supersonic bump flow

[24]

Figure 1-3: p-refinement of a quadrilateral element with hanging nodes shown in

black

In h-refinement, elements are divided into smaller elements at desired regions to

increase accuracy. Grid connectivity and number of nodes/elements are changed by

this method, which brings complications to code development. Many different

element division techniques have been developed so far. An example to h-refinement

of a two-dimensional mesh of quadrilateral elements is given in Figure 1-4 , in which

element 1 is divided into four elements and others remain the same, which results in

non-conformal elements (element 2 and 3) with hanging nodes. It is also possible to

refine quadrilateral elements without generating hanging nodes.

1

3

2

4 3

2

4

1

8

On the other hand it is easier to divide triangular elements without creating any

hanging nodes as seen in Figure 1-5. The element division shown in Figure 1-5 is not

a preferred method because dividing an element several times results in very skew

elements in the region where high accuracy is needed. Another method introduced by

Rivara considers the longest edge of the element to be refined [29]. In this algorithm,

elements selected for refinement and their neighboring elements are bisected by their

longest edges until the mesh is conforming.. This is an effective method to refine

mesh without causing mesh degeneracy. But in this method mesh outside the desired

refinement region is also refined to satisfy the longest edge rule. Suarez et al. pointed

out this undesired mesh propagation problem [30], an extreme case of which is

illustrated in Figure 1-6.

Figure 1-4: h-refinement of a quadrilateral element with hanging nodes shown in

black

Figure 1-5: h-refinement of a triangular element without hanging nodes

1

3

2

4 3

2

4

9

Figure 1-6: Demonstration of undesired mesh propagation problem of the longest edge refinement algorithm [30]

1.4 Current Study

In the current study, a solver based on LSFEM with an h-type AMR technique is

developed to numerically simulate 2-D planar and axisymmetric flows governed by

the compressible Euler equations. The developed solver works with both

quadrilateral and triangular elements of linear and quadratic shape functions and it is

tested with transonic and supersonic flow problems. Numerical results are in good

agreement with the literature for both element types and orders. The developed code

solves unsteady versions of the governing equations and marches in time to obtain a

steady state solution, if one exists. In LSFEM, numerically introduced artificial

viscosity can be decreased by using smaller time steps in order to sharply resolve

shock waves. But using a small time step on a relatively coarse mesh causes

oscillations near the shock region. In order to achieve sharp shock resolution without

oscillations, AMR technique is employed for meshes with triangular elements.

It is also worth to mention that the current solver is developed starting from an initial

code which was capable of solving 2-D planar, steady, incompressible Navier-Stokes

equations in the � − � − � formulation on linear quadrilateral mesh. The code is

developed in C++ language.

10

CHAPTER 2

GOVERNING EQUATIONS & LEAST SQUARES FEM FORMULATION

In this chapter, first the governing equations for 2-D, compressible, inviscid flow will

be recalled. Time-discretization of these equations will be done and the resultant

equations will be linearized. Finally least-squares finite element formulation of the

resultant equation set will be provided.

2.1 Governing Equations

Fundamental equations of fluid mechanics are the conservation of mass, momentum

and energy. For an inviscid compressible flow, governing equations are the following

��� + ∇ ∙ �!"#$ = 0 (2.1)

�!"#� + !"# ∙ ∇!"# + 1� ∇� = 0 (2.2)

��� + !"# ∙ ∇� + %� ∇ ∙ !"#$ = 0 (2.3)

where � is the fluid density, !"# is the velocity vector, � is the pressure and the % is the

constant specific heat ratio. Conservation of energy is written in terms of pressure by

using the perfect gas relation. In this thesis, compressible Euler equations are solved

for two spatial dimensions, both planar and axisymmetric. Conservation equations

for a planar flow in the �& Cartesian plane are given below

11

��� + � ���� + � ���� + � �'�& + ' ���& = 0 (2.4)

��� + � ���� + ' ���& + 1� ���� = 0 (2.5)

�'� + � �'�� + ' �'�& + 1� ���& = 0 (2.6)

��� + %� ���� + � ���� + %� �'�& + ' ���& = 0 (2.7)

where � and ' are the velocity components in � and & directions.

For an axisymmetric flow, variables are represented in cylindrical coordinates; (, *

and + with the following assumptions; unknowns (�, '- , '. , �) are independent of the

angular variable * and the velocity in the * direction is zero. Conservation equations

for an axisymmetric flow in the (+ plane are given below

��� + � �'-�+ + '- ���+ + � �'.�( + '. ���( + �'.( = 0 (2.8)

�'-� + '- �'-�+ + '. �'-�( + 1� ���+ = 0 (2.9)

�'.� + '- �'.�+ + '. �'.�( + 1� ���( = 0 (2.10)

��� + %� �'-�+ + '- ���+ + %� �'.�( + '. ���( + % �'.( = 0 (2.11)

where '- and '. are the velocity components in + and ( directions, respectively.

Conservation equations for planar flows (2.4-2.7) can be written in matrix form in

terms of primitive variables (in non-conservative form) as follows

�0� + 1� �0�� + 1� �0�& = 0 (2.12)

12

where 02 = (�, �, ', �) is the unknown vector and the entries of the 1� and 1�

matrices are given below

1� = 3� � 0 00 � 0 1 �⁄0 0 � 00 %� 0 � 5

1� = 3' 0 � 00 ' 0 00 0 ' 1 �⁄0 0 %� ' 5 (2.13)

Time discretization of the equation set (2.12) is done by the backward Euler method,

during which a new unknown vector is introduced as ∆0 = 0�7� − 0�, where the

superscripts denote discrete time levels. This relation can also be expressed as

follows

3��'�5�7� = 3��'�5� + 3∆�∆�∆'∆�5

(2.14)

Continuity equation (2.4), written for the current time level 8 + 1 can be expressed

in terms of the new ∆ unknowns and known values from the previous time level 8 as

follows

Δ�Δ + (∆� + ��) �(∆� + ��)�� + (∆� + ��) �(∆� + ��)��+ (∆� + ��) �(∆' + '�)�& + (∆' + '�) �(∆� + ��)�& = 0

(2.15)

Neglecting the following higher order terms in the above equation

∆� �∆��� , ∆� �∆��� , ∆' �∆��& , ∆� �∆'�&

new time-step values

previous time-step values (known)

new introduced unknown vector

13

and re-arranging it such that known values are on the right-hand side, following

continuity equation is obtained

Δ�Δ + Δ� ����� + �� �∆��� + ∆� ����� + �� �Δ��� + Δ� �'��& + �� �∆'�&+ ∆' ����& + '� �Δ��& = − :�� ����� + �� ����� + �� �'��& + '� ����& ;

(2.16)

Similar to the continuity equation, momentum equation in the �-direction (2.5)

written at time level 8 + 1 can be expressed in terms of the new Δ unknowns as

follows

Δ�Δ + (∆� + ��) �(∆� + ��)�� + (∆' + '�) �(∆� + ��)�&+ 1∆� + �� �(∆� + ��)�� = 0

(2.17)

Pressure term can be simplified by multiplying its numerator and denominator by (�� − Δ�). Neglecting higher order terms and re-arranging equation (2.17), �-

momentum equation becomes

Δ�Δ + Δ� ����� + �� �Δ��� + Δ' ����& + '� �Δ��& − Δ� 1(��)� �����+ 1�� �∆��� = − :�� ����� + '� ����& + 1�� ����� ;

(2.18)

Similarly, y-momentum equation can be obtained as

Δ'Δ + Δ� �'8�� + �8 �Δ'�� + Δ' �'8�& + '8 �Δ'�& − Δ� 1(�8)2 ��8�& + 1�8 �∆��&= − =�8 �'8�� + '8 �'8�& + 1�8 ��8�& >

(2.19)

14

Following a similar procedure energy equation (2.7) in terms of the new Δ unknowns

can be written as

Δ�Δ� + %∆� ��8�� + %�8 �∆��� + ∆� ��8�� + �8 �∆��� + %∆� �'8�& + %�8 �∆'�&+ ∆' ��8�& + '8 �∆��& = − =%�8 ��8�� + �8 ��8�� + %�8 �'8�& + '8 ��8�& >

(2.20)

To summarize, equation set (2.4-2.7) is now discretized in time and linearized,

yielding the following set of equations, for which the unknown vector is ∆02 = (∆�, ∆�, ∆', ∆�).

Δ�Δ + Δ� ����� + �� �∆��� + ∆� ����� + �� �Δ��� + Δ� �'��& + �� �∆'�&+ ∆' ����& + '� �Δ��& = − :�� ����� + �� ����� + �� �'��& + '� ����& ;

(2.21)

Δ�Δ + Δ� ��8�� + �8 �Δ��� + Δ' ��8�& + '8 �Δ��& − Δ� 1(�8)2 ��8�� + 1�8 �∆���= − =�8 ��8�� + '8 ��8�& + 1�8 ��8�� >

(2.22)

Δ'Δ + Δ� �'8�� + �8 �Δ'�� + Δ' �'8�& + '8 �Δ'�& − Δ� 1(�8)2 ��8�& + 1�8 �∆��&= − =�8 �'8�� + '8 �'8�& + 1�8 ��8�& >

(2.23)

Δ�Δ� + %∆� ��8�� + %�8 �∆��� + ∆� ��8�� + �8 �∆��� + %∆� �'8�& + %�8 �∆'�&+ ∆' ��8�& + '8 �∆��& = − =%�8 ��8�� + �8 ��8�� + %�8 �'8�& + '8 ��8�& >

(2.24)

Using the matrix notation previously used in equation (2.12), new set of equations

can be written as

1∆0 = 1�� �∆0�� + 1�� �∆0�& + 1@�∆0 = A (2.25)

15

where 1� and 1� matrices are the same as given before in equation (2.13) and the

newly introduced 1@ matrix and A vector are given as follows (from now on, the

superscript "8" which denotes the previous time-step in the unknowns will be

omitted for clarity)

1@ =BCCCCCCCD

1∆ + ���� + �'�& ���� ���& 0− 1�� ���� 1∆ + ���� ���& 0− 1�� ���& �'�& 1∆ + �'�& 0

0 ���� ���& 1∆ + % :���� + �'�&;EFFFFFFFG (2.26)

A = −BCCCCCCCD � ���� + � ���� + � �'�& + ' ����� ���� + ' ���& + 1� ����� �'�� + ' �'�& + 1� ���&%� ���� + � ���� + %� �'�& + ' ���&EFF

FFFFFG (2.27)

Following the same procedure with the planar case, for axisymmetric problems it can

be shown that only the 1@ matrix and the A vector are modified as follows

1@HI� = 1@ +BCCCCD'.( 0 �( 00 0 0 00 0 0 00 0 %�( %'.( EFF

FFG , AHI� = A −BCCCCD

�'.(00%�'.( EFFFFG (2.28)

and terms �, ', � and & in 1�, 1�, 1@ and A are expressed as '-, '., z and (,

respectively.

16

Calculations in the code are performed with the non-dimensional variables. Non-

dimensionalization of these variables are done as follows

�∗ = ��K �∗ = �LK '∗ = 'LK �∗ = ��KLK�

where the subscript "∞" denotes the free-stream values and "L" is the speed of

sound.

2.2 Least-Squares Finite Element Formulation

Residual of equation set (2.21-2.24) is defined as follows

N = 1Δ0 − A (2.29)

LSFEM is based on the minimization of this residual function over the problem

domain, resulting in the minimization of the following functional [19]

O(Δ0) = 12 P N2NQΩ S (2.30)

In FEM, the domain is divided into finite number of non-overlapping subdomains,

which are called elements and the variation of the unknowns over these elements are

approximated by the use of nodal values of the unknowns and polynomial shape

functions as given below [19]

Δ0 ≈ U VW∆0WXYXWZ� (2.31)

17

where NEN is the number of elemental nodes, ∆0W are the nodal unknown values that

need to be calculated and VW are known shape functions of space coordinates. More

details about the shape functions used in this study will be given in the next section.

Substituting the above approximate representation of the unknowns into equation

(2.30) written over a single element, and minimizing the functional in a least squares

sense it is possible to get the following elemental linear algebraic equation set

[�∆0� = \� (2.32)

where ∆0� is the unknown variable vector, [� and \� are the elemental stiffness

matrix and force vector of an arbitrary element ] given as follows

[�W� = P(1V�)2 1VW$QΩ S^ (2.33)

\�� = P(1V�)2AQ٠S^ (2.34)

where the differential operator 1 is

1 = 1� ��� + 1� ��& + 1@ (2.35)

In the developed code, elemental stiffness matrix [� and force vector \� are

numerically calculated using Gauss-Legendre quadrature [31]. After the calculation

of all elemental systems, they are assembled into a global set of equations. As an

important property of LSFEM, global stiffness matrix turns out to be symmetric

positive-definite, which is solved using an efficient element-by-element Jacobi

preconditioned Conjugate Gradient method [31].

Although the formulation presented above involves unsteady terms, in this study

only steady problems are solved through a time marching procedure. Upon

18

convergence of a solution to steady-state unknown vector ∆0 is expected to approach

zero. Based on this, a convergence criterion is defined as the node number averaged �� norm of the unknowns as shown below, which is calculated for (��, ��, �', ��)

separately

_U(Δ0��)�NN�Z� NNa b = 1,4 (2.36)

where NN is the total number of nodes in the mesh.

2.3 Lagrange Shape Functions

In this thesis, 2-D triangular and quadrilateral elements in linear and quadratic forms

are used to approximate the solution. Linear (4-node) and quadratic (9-node)

quadrilateral master elements are shown in Figure 2-1.

Figure 2-1: 4-node linear and 9-node quadratic quadrilateral master elements

Shape functions of the linear quadrilateral element are

V� = 14 (1 − d)(1 − e) V� = 14 (1 + d)(1 − e)

Vf = 14 (1 + d)(1 + e) Vg = 14 (1 − d)(1 + e)

(2.37)

ξ

η

1 2

3 4

5

6

7

8 9

ξ

η

1 2

3 4

19

and the ones for the 9-node quadrilateral are

V� = 14 (d� − d)(e� − e)

V� = 14 (d� + d)(e� − e)

(2.38)

Vf = 14 (d� + d)(e� + e)

Vg = 14 (d� − d)(e� + e) Vh = 12 (1 − d�)(e� − e)

Vi = 12 (d� + d)(1 − e�)

Vj = 12 (1 − d�)(e� + e)

Vk = 12 (d� − d)(1 − e�) Vl = (1 − d�)(1 − e�)

Linear (3-node) and quadratic (6-node) triangular master elements are shown in

Figure 2-2.

Figure 2-2: 3-node linear and 6-node quadratic triangular master elements

Shape functions for the 3-node triangular elements are

V� = 1 − d − e

V� = d Vf = e (2.39)

and the ones for the 6-node triangular element are given as follows

ξ

η

1 2

3

(0,1)

(1,0) (0,0)

ξ

η

1 2

3

(0,1)

(1,0) (0,0) 4

5 6

20

V� = 2(1 − d − e) :12 − d − e;

V� = 2d :d − 12;

(2.40) Vf = 2e :e − 12;

Vg = 4(1 − d − e)d Vh = 4de

Vi = 4(1 − d − e)e

When the shape functions are defined over master elements as given above, integrals

of the elemental stiffness matrices and force vectors require a change of variables

from (�, &) to (d, e), for which the standard Jacobian transformation is used [31].

2.4 Boundary Conditions

In order to have a well-posed problem, boundary conditions, number of which

depends on the problem physics, have to be specified. For the equation system solved

in this study, boundary conditions are easily imposed as Dirichlet type except for the

wall boundary conditions. For inviscid flows, flow at the wall boundaries is tangent

to the wall. Flow tangency boundary condition is usually imposed by using the

penalty method or the coordinate rotation method. Both methods are implemented in

the developed code, and details about them are presented below.

2.4.1 Penalty Method

In the penalty method, flow-tangency boundary condition is imposed by adding the

following functional to equation (2.30)

m2 P !"# ∙ 8"#$�QΓ opqrr

= m2 P ��� Qà opqrr

(2.41)

where !"# is the velocity vector , 8"# is the unit vector normal to the wall boundary and m is the penalty parameter. Velocity components at the wall boundary are shown in

21

Figure 2-3. Penalty parameter determines how strongly the flow tangency boundary

condition is imposed. Selecting a too large penalty parameter results in slower

convergence, whereas a too small penalty parameter value decreases the accuracy at

the wall boundaries. In this study, penalty parameter is selected in the range 0.05 and

0.2.

Figure 2-3: Velocity components at a solid wall

The angle * at Figure 2-3 is calculated at the nodes and more than one element may

share the same wall boundary node as seen in Figure 2-4

Figure 2-4: Wall boundary node A

For such cases the angle at the wall boundary node A can be calculated as follows,

by using the angles and lengths of the adjacent edges [24]

*s = t�*� + t�*�t� + t� (2.42)

A

θ1

θ2

l1

l2

u

v

ut

un

x

y

θ

22

Knowing these, velocity normal to the wall boundary can be written as

�� = − sin(*) � + cos(*) ' (2.43)

where * is the angle between wall boundary and the �-coordinate. Considering the

velocity components in equation (2.43) to be written for time level 8 + 1 and

expanding them in terms of velocities at the previous time level 8 and the velocity

differences between two time levels yields

�� = − sin(*) Δ� + cos(*)Δ' − sin(*) �� + cos(*)'� (2.44)

The last two terms in the above equation are known values from the previous time

step. Introducing equation (2.44) into the functional (2.41), elemental stiffness matrix [� and force vector \� are modified at the wall boundary nodes by adding following

matrix and vector respectively

[�z��H��{ = [� + m 30 0 0 00 sin�(*) − sin(*) cos(*) 00 − sin(*) cos(*) cos�(*) 00 0 0 05 (2.45)

\�z��H��{ = \� + m 3 0(sin(*) � − cos(*) ') sin(*)(sin(*) � − cos(*) ') cos(*)0 5 (2.46)

Note that elemental stiffness matrix and force vector are modified at the wall

boundary elements' wall boundary nodes only. It is also worth to mention that this

modification do not destroy the important symmetric positive definite property of the

discretized equation system.

23

2.4.2 Coordinate Rotation Method

Coordinate rotation method is a direct approach to impose flow tangency at wall

boundaries. In this method, wall boundary nodes are rotated to a local coordinate

system in which the velocity components are tangent and normal to the wall as seen

in Figure 2-3. After this rotation, boundary condition can be implemented as a

Dirichlet boundary condition by setting the normal velocity component to zero. The

relation between tangent and normal velocity components and the original velocity

components is given below

|����} = ~ cos (*) sin (*)−sin (*) cos (*)� |�'} (2.47)

Considering that this coordinate transformation only affects the velocity components

but not density or pressure, the following transformation matrix between the

unknowns of the original and rotated local coordinates can be written

� = 31 0 0 00 cos(*) sin(*) 00 − sin(*) cos(*) 00 0 0 15 (2.48)

In order to set the normal velocity components at the wall boundaries to zero

elemental system for the wall boundary elements are modified using the

transformation matrix as follows

(�2[��)∆0 = (�2\�) (2.49)

Similar to the penalty approach, coordinate rotation also preserves the symmetric

positive-definite property of the final equation set.

24

2.5 Adaptive Mesh Refinement (AMR)

In this study AMR is mainly used for efficient and accurate shock capturing and it is

used only with triangular meshes. For this purpose first an error indicator is

calculated for each element and the elements with an error exceeding the user

defined tolerance value are divided into smaller ones. Details of these two steps are

given below.

2.5.1 Error Indicator

The first step of grid adaptation is assigning error values for each element to

determine the elements that need to be refined. These errors can be estimated by

many different approaches. In this thesis the gradient approach, which is very

suitable for supersonic inviscid flows, is used [38]. It uses the following area

weighted �� norm of the pressure gradient as an error indicator

](( = 1����|∇�| (2.50)

where 1���� is the area of the element. In determining the element division, error

indicators are scaled with the average error indicator over the problem domain,

which makes the algorithm more robust and problem independent. Note that

although pressure gradient is used in error calculation in this study, one can also

adopt other variables for this purpose.

2.5.2 Element Division

After determining the elements that need to be refined, element division is done by

paying attention to preserving the conformity of the mesh. Element division

algorithm used in this study is based on the bisection of the element by its longest

edge [29]. In this algorithm, elements which are flagged for division are divided by

25

their longest edges and then the non-conforming neighboring elements are bisected

across their non-conforming edges to get rid of the undesired hanging nodes. Note

that after the first division non-conforming neighbors are not recursively divided by

their longest edges until the mesh is conforming as in Rivara's method [29].

A simple illustration of element division is shown in Figure 2-7. If the top element is

flagged to be refined, first it is divided by its longest edge and then the neighbor

element is divided to make the mesh conformal. As seen in the figure the

neighboring element is not necessarily divided through its longest edge.

Figure 2-5: Element division

After the element division process is done, mesh connectivity is updated and new

coordinates, flow-field variables and boundary condition information are assigned to

the newly created nodes. This algorithm is not the best alternative if we strictly

consider avoiding grid degeneracy, but it is simple and FEM is known to tolerate grid

skewness to a certain degree. In addition, possible propagation problem

demonstrated in Figure 1-6, which may occur in the recursively longest-edge

refinement algorithms is avoided.

element flagged to be refined

will be divided for conformity

26

CHAPTER 3

RESULTS

In this chapter, results obtained with the developed code for various problems are

given. First, supersonic and transonic flow over a circular bump is simulated and the

results are compared with the ones found in literature. For this first problem both

linear and quadratic types of quadrilateral and triangular elements are used. For the

triangular elements, transonic and supersonic cases are also solved with AMR. The

second test problem is chosen to be the supersonic flow over a wedge, which has a

known analytical solution. In this problem AMR algorithm is challenged by starting

the solution from a very coarse initial mesh. Wedge problem is also used to study the

effect of time step on numerical dissipation and shock capturing. Then flow over an

axisymmetric body is simulated and results are compared with available data in

literature. The final test case is the supersonic flow in a scramjet inlet.

3.1 Supersonic Flow Over a Circular Bump

In this problem supersonic flow inside a channel with 4 % blockage by a circular

bump as seen in the Figure 3-1 is considered. Channel length and height is 3.0 and

1.0 units respectively. Circular bump is positioned at the lower wall with a height of

0.04 units. In this classical benchmark problem it is known that as the supersonic

flow with the inlet Mach number of 1.65 interacts with the bump an oblique shock

emanates from the leading edge of the bump and reflects from the upper wall, which

27

further interacts with the oblique shock that originates from the trailing edge of the

bump.

The inlet boundary condition of the problem is given in the Figure 3-1. The specific

heat ratio of the flowing gas is taken as 1.4. The exit boundary is left free (nothing

special is done for it in the code) and on the upper and lower walls, flow tangency

boundary condition is imposed by the penalty method.

Figure 3-1: Sketch of the supersonic flow over a circular bump problem

For this first test case calculations are done for both quadrilateral and triangular

meshes with linear and quadratic interpolating functions and results will be compared

with the data found in the literature [33]. Quadrilateral and triangular meshes should

be similar for a proper comparison of the solutions obtained with them. For this

purpose, first the mesh of quadrilateral elements shown in Figure 3-2 is created and

the triangular mesh, also shown in Figure 3-2 is obtained by dividing each

quadrilateral element into two, ending up with the same number of nodes and

unknowns for each mesh. For both meshes, on the bottom wall there are 30

uniformly distributed elements on the bump and after the bump and there are 20

elements before the bump clustered to the bump. On the inlet and exit there are 30

uniformly distributed elements.

ρ = 1.0 u = 1.65 v = 0.0 p = 0.7143

x

y

1.0

1.0

1.0

28

Figure 3-2: Quadrilateral and triangular computational meshes used for supersonic bump problem (2511 nodes)

As clearly stated by Jiang [19] selection of the number of Gauss quadrature points is

important when least-squares formulation is used. Number of quadrature points

(NGP) is selected as 4 for quadratic and linear quadrilateral elements and for linear

triangular elements. However, due to stability issues NGP is selected as 3 for

quadratic triangular elements.

Time step is set to 0.1 for the solutions of the supersonic bump problem. The

numerical viscosity produced by the least-squares formulation is a function of time

step, of which details are given by Jiang et al. [23]. Selecting a smaller time-step

decreases numerical viscosity and as a result shock resolution increases, but it may

also cause oscillations close to the discontinuities. The effect of time step will be

inspected in detail for the flow over wedge problem.

Pressure contours of the flow field for both meshes and both element orders are

given in Figure 3-3. Interactions of shock waves stated in the problem description are

clearly seen in the figure. For quadrilateral and triangular linear elements (Figure 3-3

a, c) shock waves are relatively dispersed. Shock resolution improves when second

order elements are used, which is expected because the number of nodes almost

doubles as compared to the use of linear elements.

29

Figure 3-3: Pressure contours for supersonic bump problem with ������ = 1.65 and � = 0.1: (a) linear quadrilateral elements (b) quadratic quadrilateral elements (c) linear triangular elements (d) quadratic triangular elements

0.5 0.6 0.7 0.8 0.9 1 1.1

a)

b)

c)

d)

�:

30

Before comparing these results in further detail, let us apply the AMR technique to

the mesh of linear triangular elements. The mesh after 6 adaptation cycles can be

seen in Figure 3-4. For this adaptive solution first the original triangular mesh of

linear elements is used without mesh adaptation for 40 time steps. Then at every 10

iteration mesh is adapted with a pre-defined maximum adaptation level and

minimum element size. After mesh is adapted 6 times, iterations continue until the

convergence criterion is satisfied. It is clearly seen in Figure 3-4 how AMR increases

the mesh resolution at the shock regions. A closer look at the adapted mesh around

the leading edge shock is given in Figure 3-5, for which black and red colors indicate

initial and final meshes, respectively.

Figure 3-4: Final adapted mesh for supersonic bump problem

Figure 3-5: Zoomed view of the adapted mesh around the leading edge shock of supersonic bump problem

x

y

1.55 1.6 1.65 1.7

0.6

0.65

0.7

31

Pressure contours obtained with the final adapted mesh are given in Figure 3-6.

Figure 3-6: Pressure contours of the final adapted solution of the supersonic bump problem

As seen in Figure 3-6, compared to previous non-adaptive solutions shocks are

captured sharply without any oscillations in the flow field. A 3-D view of the

pressure contour plot is also presented in Figure 3-7. In order to compare shock

resolutions better, slices are taken at & = 0.5 from quadrilateral and triangular

meshes with order � = 1 (linear) and � = 2 (quadratic) and also from adapted

triangular mesh with � = 1 (see Figure 3-7) and 5 different solutions are compared

in Figure 3-8.

Figure 3-7: Pressure data extraction from y=0.5 for the supersonic bump problem

0.5 0.6 0.7 0.8 0.9 1 1.1�:

32

As seen in Figure 3-7, the slice at the & = 0.5 intersects the leading edge shock wave

and its reflection from the upper wall, as well as the trailing edge shock wave after its

interaction with the reflected shock. All of these flow features can also be seen in

Figure 3-8.

Figure 3-8: Comparison of pressure distribution at y=0.5 for 5 different solutions of the supersonic bump problem

Figure 3-8 illustrates the improvement in the shock resolution with AMR. With

linear elements, for both quadrilateral and triangular meshes, shock resolution is

poor. Increasing element order, improves the shock resolution to a certain degree, but

cannot capture discontinuities as good as the adapted solution.

For further comparison, Mach number distribution at the lower wall is presented in

Figure 3-9. The reason of the deviations can be found by analyzing the pressure

contours given in Figures 3-3 and 3-6. In these contour plots, it is seen that the shock

wave reflecting from the upper wall leaves from the supersonic exit. When the shock

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.5 1 1.5 2 2.5 3

p

x

quad, p=1

quad, p=2

tri, p=1

tri, p=2

tri, adapt.

33

resolution is poor, shock wave is smeared and this affects the lower wall near the

supersonic exit. Also, there is a shift in the Mach number distribution at the right side

of the bump, which is only seen in the neighboring elements to the wall boundary.

This effect is smaller at higher order elements and it is not seen in the adapted

solution.

Figure 3-9: Mach number distribution at the lower wall of the supersonic bump problem

In Figure 3-10 Mach number distribution at the upper and lower wall are compared

with the solution of Eidelman et al. [33] and Taghaddosi et al. [24]. At the upper

wall, results are in good agreement with each other. It is seen that the reflected shock

wave at the upper wall is more sharply captured in the current solution and

Taghaddosi's solution since both results are obtained with AMR. At the lower wall,

there is also good agreement between the results except the deviation after the bump.

The Mach number at the lower wall after and before the bump is expected to be the

same for an inviscid flow, which is captured only by the current solution.

1

1.25

1.5

1.75

2

0 0.5 1 1.5 2 2.5 3

Mach Number

x

quad, p=1

quad, p=2

tri, p=1

tri, p=2

tri, adapt.

34

Figure 3-10: Mach number distribution at the upper (top) and lower wall (bottom) for the supersonic bump problem

Convergence histories of all the performed solutions are shown in Figure 3-11. As

seen the convergence rate of the solution with quadratic triangular element after 60

iterations is significantly slower compared to the other solutions. In the adapted

1

1.25

1.5

1.75

2

0 0.5 1 1.5 2 2.5 3

Mach Number

x

adapted solution

Eidelman et al. [33]

Taghaddosi et al. [24]

1

1.25

1.5

1.75

2

0 0.5 1 1.5 2 2.5 3

Mach Number

x

35

solution, after 40 time iterations after which the flow features become apparent, mesh

adaptation takes place. At every 10 time iterations mesh adaptation is performed

without waiting for the residual to drop further and after the final adaptation 100 time

iterations was enough for the solution to converge.

Figure 3-11: Convergence histories of different solutions for the supersonic bump problem

-16

-14

-12

-10

-8

-6

-4

-2

0

0 50 100 150 200 250 300 350 400

log

(R)

Number of iterations

quad, p=1

quad, p=2

tri, p=1

tri, p=2

tri, adapt.

36

3.2 Transonic Flow Over a Circular Bump

This time, transonic flow with an inlet Mach number of 0.675 passes through a

channel with a 10 % blockage by a circular bump. Channel dimensions and the

position of the bump are the same as the previously studied problem. Flow enters the

channel at a subsonic speed and reaches to supersonic speeds on the bump, which

results in the formation of a shock wave.

The inlet boundary condition of the problem is given in the Figure 3-12. The specific

heat ratio, %, is taken as 1.4. At the exit boundary, pressure is set to inlet pressure

value of 0.7143. On the upper and lower walls, flow tangency boundary condition is

imposed. For this problem, calculations are be done with both quadrilateral and

triangular meshes with linear and quadratic interpolating functions, similar to the

previous problem, and the results are compared with the data found in the literature

[24, 33].

Figure 3-12: Sketch of the transonic flow over a circular bump problem

Similar to the previous problem first the quadrilateral mesh seen in Figure 3-13 is

prepared and the triangular mesh is obtained by dividing each quadrilateral element

into 2. For each mesh, the bottom wall has 40 uniformly distributed elements on the

bump and 20 elements on the both sides of the bump clustered to the bump. There

are 30 uniformly distributed elements across the channel height. Both meshes have

1.0

1.0

� = 1.0 � = 0.675 ' = 0.0 � = 0.7143 x

y � = 0.7143

37

2511 nodes for the linear elements. Number of Gauss quadrature points and time step

selections are the same as the previous supersonic bump problem.

Figure 3-13: Computational meshes of (top) rectangular and (bottom) triangular elements, both having 2511 nodes

Mach number contours of the flow-field for both meshes and both element orders are

given in Figure 3-14. It is seen from the figure that there is a viscous effect after the

shock wave which is also mentioned in Taghaddosi et al. [24]. This effect is greatest

for the use of quadrilateral linear elements. Also for second order elements,

oscillatory behavior can be seen in the flow field, especially after the shock wave. In

all of the solutions, shock wave is captured sharply and the shock location is

approximately at � = 1.72 on the bump, which is in good agreement with the

literature data.

38

Figure 3-14: Mach number contours for transonic bump problem with ������ = 0.675 (a) linear quadrilateral elements (b) quadratic quadrilateral elements (c) linear triangular elements (d) quadratic triangular elements

0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25

a)

b)

c)

d)

�:

39

Mach number distribution at the lower wall is given in Figure 3-15. Results are in

good agreement with the literature data except small shifts in the Mach number after

the bump, which is related to the viscous effect mentioned in the previous paragraph.

This effect is higher for quadrilateral elements and lower for triangular elements of

which results agree well with Ni [34]. Taghaddosi et al. [24] mentioned in their paper

about this effect and compared their results with Eidelman et al. [33] and Ni [34]. In

this comparison they observed a similar deviation of Mach number, which amplifies

by the use of larger time steps [24].

Figure 3-15: Mach number distribution at the lower wall for different element types and orders

Before investigating the results further, solution with AMR will be presented. Unlike

the previous problem, AMR is done with a coarser mesh. The generated mesh seen in

figure 3-16 consists of 797 nodes which is approximately three times less than the

previous mesh. Adaptive computations with this coarse mesh are done with a time

step of 0.2. Adaptation took place at every 20 time iterations until reaching 8 levels

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Mach Number

x

quad, p=1

quad, p=2

tri, p=1

tri, p=2

Ni [34]

Taghaddosi et al. [24]

40

of adaptation. Zoomed view of initial and final adapted mesh at the shock position on

the bump is given in Figure 3-17, which illustrates the improvement in the grid

resolution at the shock wave. The modified grid and pressure contours near bump at

different adaptation levels are given in Figure 3-18 which shows the improvement in

shock resolution.

Figure 3-16: Initial coarse mesh for AMR

Figure 3-17: Initial (black) and adapted (red) mesh for the transonic bump problem

x

y

1.7 1.72 1.74 1.76

0.08

0.1

0.12

41

Figure 3-18: Initial and adapted meshes and pressure contours near bump, � = 0.2:

(a) Initial mesh, (b) 2nd adaptation, (c) 4th adaptation, (d) 8th adaptation

0.33 0.45 0.57 0.69 0.81

a)

b)

c)

d)

42

In Figure 3-19, Mach number distribution over the lower wall obtained with the

adapted solution is compared with the literature data. Adapted solution results agree

very well with the results of Eidelman et al. [33] and Ni [34] throughout the channel

wall and with Taghaddosi et al. [24] before the trailing edge. After the trailing edge,

results of Taghaddosi et al. under predict the Mach number as the authors stated.

They used a coarse quadrilateral initial mesh and employed r-type refinement and

after the adaptation grid became even coarser at the downstream of the bump, which

apparently increased the viscous effect. Similar behavior is also observed in the

current study, especially with non-adaptive solutions as discussed before. In order to

see this effect more clearly, Mach number distribution at the exit plane is given in

Figure 3-20. It is seen that underestimation of Mach number at the lower wall due to

viscous effect is limited to the elements right above the wall boundary only.

Figure 3-19: Mach number distribution at the lower wall of adapted solution of the transonic bump problem

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Mach Number

x

adapted solution

Eidelman et al. [33]

Taghaddosi et al. [24]

Ni [34]

43

Figure 3-20: Mach number distribution at the exit of the transonic bump problem

Convergence history of the solutions are given in the below graph.

Figure 3-21: Convergence histories of different solutions for the supersonic bump problem

0

0.1

0.2

0.3

0.4

0.5

0.4 0.5 0.6 0.7 0.8

y

Mach number

quad, p=1

quad, p=2

tri, p=1

tri, p=2

tri, adapt

-14

-12

-10

-8

-6

-4

-2

0

0 100 200 300 400 500 600 700 800 900 1000

log

(R)

Number of iterations

quad, p=1

quad, p=2

tri, p=1

tri, p=2

tri, adapt.

44

3.3 Supersonic Ramp Flow

In this test case, supersonic flow with an inlet Mach number of 2.0 flows over a

wedge which makes a 5° angle with the free stream flow as seen in Figure 3-22. Inlet

boundary conditions are given in the figure, flow tangency boundary condition is

imposed at the lower and upper walls and the right boundary is left free. Gas enters

the domain through the inlet at the left and an oblique shock wave emanates from the

leading edge of the ramp with an angle of 34.3°. Flow properties after the shock

wave can be calculated analytically. In this problem, first the effect of time step in

LSFEM formulation will be inspected and then LSFEM with AMR technique will be

tested by solving the problem with a very coarse initial mesh.

Figure 3-22: Sketch of the supersonic ramp flow

Numerical viscosity produced by the least-squares formulation is known to be a

function of time step [23]. Using a smaller time step decreases numerical viscosity

and as a result shock resolution increases. However, it may also cause oscillations in

the solution. In order to see the effect of time step on numerical viscosity and shock

resolution, triangular mesh given in Figure 3-23 is generated.

x

y

1 3

3

5°

� = 1.4 � = 2.0 ' = 0.0 � = 1.0

45

Figure 3-23: Uniform mesh for the supersonic ramp flow (7512 elements, 3872 nodes)

Pressure contours for time step values of 0.3, 0.1, 0.05 and 0.02 are given in Figure

3-24. The effect of time step on shock resolution can be seen in this figure. As

expected, as the time step decreases shock wave becomes thinner. However, by

decreasing the time step we also observe oscillations near the shock region. In order

to see this behavior clearly, pressure data at � = 2.8 is presented in Figure 3-25.

Shock wave intersects the � = 2.8 line approximately at & = 1.226.

As seen in Figure 3-25 as the time step gets smaller shock resolution gets better, but

overshoots and undershoots before and after the shock are amplified, clearly showing

the link between the time step and numerical dissipation. It is also observed that

decreasing time step results in a small shift of the shock position. The possible reason

of this is the oscillations before the shock wave. Another possible reason is stated in

the literature as using non-conservative form of the governing equations [28, 35].

Further discussion of this issue will be done after solving the problem with AMR.

46

Figure 3-24: Pressure contours of supersonic ramp flow obtained with different �

Figure 3-25: Effect of time step on pressure distribution at x=2.8 for supersonic ramp

flow

p: 1.00 1.04 1.08 1.12 1.16 1.20 1.24 1.28 1.32

0.9

1

1.1

1.2

1.3

1.4

0 0.5 1 1.5 2 2.5 3

p

y

∆t = 0.3

∆t = 0.1

∆t = 0.05

∆t = 0.02

exact solution

∆t = 0.3 ∆t = 0.1

∆t = 0.05 ∆t = 0.02

47

It is seen that, the uniform grid used for this problem (Figure 3-23) is not fine enough

to capture the shock wave accurately without oscillations. AMR can be used for

better shock capturing. To show the real power of AMR, a very coarse initial mesh of

113 nodes and 194 elements (approximately 40 times coarser than the previously

used uniform grid) is generated (Figure 3-26).

Figure 3-26: Coarse initial mesh for solving the supersonic ramp problem with AMR (113 nodes and 194 elements)

Adaptive solution of the problem is started with a time step of 0.2 and after 7

adaptation cycles, time step is reduced to 0.1 and 2 more mesh adaptation is done

before finalizing the solution. Figure 3-27 shows the modified grid and improvement

in shock resolution at different adaptation levels. After the final adaptation, number

of elements and nodes become 11303 and 5684, respectively. It can be seen in Figure

3-29 that only the elements at the shock location are refined and undesired mesh

propagation is avoided.

Because the solution started from a very coarse initial mesh, the final mesh contains

very skew elements near the shock region, as seen in Figure 3-29. In order to see the

effect of mesh skewness on the solution, a new simulation is done with a modified

AMR algorithm, in which skew elements are also refined to make them more

orthogonal. However, it is seen that this brings no improvement to the solution, but it

only increases computational work and solution time. So it can be concluded that

LSFEM can work satisfactorily with the skew elements seen in Figure 3-29.

Figure 3-27: The grids and pressure contours for initial and adapted solutions of the supersonic ramp pro

a)

b)

c)

d)

48

The grids and pressure contours for initial and adapted solutions of the supersonic ramp problem: (a) Initial mesh, (b) 3rd adaptation, (c) 6

9th adaptation

The grids and pressure contours for initial and adapted solutions of the adaptation, (c) 6th adaptation, (d)

49

The initial and final adapted mesh is shown together in Figure 3-28 to clearly

illustrate the refinement process. Zoomed view to the refined elements at shock

position is also presented in Figure 3-29.

Figure 3-28: Initial and final adapted mesh used for the supersonic ramp problem

Figure 3-29: Zoomed view of the initial and adapted mesh at shock location

x

y

2 2.1 2.2 2.3 2.4

0.6

0.7

0.8

0.9

50

Figure 3-30 shows the pressure distribution of the adapted solution at � = 2.8.

Adapted LSFEM solution predicts the shock position correctly and provides high

shock resolution without oscillations. Comparing this result with the non-adaptive

ones (Figure 3-25), improvement in the shock resolution is quite significant. It is also

important to mention that, unlike previous observations, no deviation from the exact

shock location is observed by using the non-conservative formulation with AMR

algorithm.

Figure 3-30: Pressure distribution for adapted and exact solution at � = 2.8 for the supersonic ramp problem

Finally, convergence history of the adapted solution is given in Figure 3-31. In the

solution of the problem, adaptations are successively performed without waiting for

the residual to drop further up to the final adaptation step after which iterations

continued until the solution reaches the steady state. After 7 adaptations, time step is

reduced from 0.2 to 0.1 which is followed by 2 more adaptations.

0.9

1

1.1

1.2

1.3

1.4

0 0.5 1 1.5 2 2.5 3

p

y

adapted solution

exact solution

51

Figure 3-31: Convergence history of the adapted solution of the supersonic ramp problem

-12

-10

-8

-6

-4

-2

0

0 100 200 300 400 500 600 700

log(R)

Number of iterations

52

3.4 Flow Over an Axisymmetric Body

In this problem, supersonic flow of Mach number 3.0 over an axisymmetric body is

solved and results are compared with available data in literature [36]. The

axisymmetric body geometry is six diameters in length which has a three diameter

length secant-ogive nose with a radius of 18.88 diameters.

Figure 3-32: Axisymmetric body geometry (dimensions are in body diameters)

The boundary conditions for the freestream Mach number of 3.0 are � = 1.0, � = 3.0, ' = 0.0 and � = 0.7143. The computational domain does not include the

base of the axisymmetric body. Flow tangency boundary condition imposed at the

axisymmetric body and top and right boundaries are left free. Initial mesh around the

body is given in the Figure 3-33. Time step is set to 0.05 for the solution of the

problem. The mesh is adapted to get the final results. Adapted mesh and pressure

contours can be seen in the Figure 3-34.

Figure 3-33: Initial mesh around the axisymmetric body with 4092 nodes

7°

3 2 1

53

As seen in 3-34, oblique shock wave attached to the nose of the body is captured

well. Expansion waves at the end of the nose and at the boattail can also be seen.

Figure 3-34: Adapted mesh and pressure contours of the axisymmetric flow problem

In Figure 3-35, surface pressure ratio is compared with the one found in literature

and good agreement is observed.

p: 0.4 0.55 0.7 0.85 1 1.15 1.3

54

Figure 3-35: Pressure distribution on the axisymmetric body

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6

p/p

∞∞ ∞∞

x

adapted solution

Schiff & Sturek [36]

55

3.5 Flow Through Scramjet Inlet

In this final test case, supersonic flow of Mach number 2.0 enters the scramjet inlet

and compressed through a 10° ramp as seen in Figure 3-36. Inlet boundary

conditions are also given in figure. Flow tangency boundary condition is imposed at

the solid wall of the inlet by penalty method and only the upper half of the flow

domain is studied by setting the velocity in the vertical direction to zero at the

symmetry line. Supersonic exit is left free.

Figure 3-36: Sketch of the inlet problem (all dimensions are in meters)

Initial grid consists of 1422 triangular elements and 775 nodes and can be seen in

Figure 3-37. Solution started with a time step of 0.1 and after 8 adaptations,

numerical viscosity reduced by setting the time step to 0.05 followed by 2 more

adaptations which can also be seen in Figure 3-40 from the jumps of the residuals.

Figure 3-37: Initial mesh for the scramjet inlet problem with 1422 elements and 775 nodes

� = 2.0 � = 26.5 ��L � = 226[ 10°

1 1 3

2

56

In Figure 3-38 adapted grid and pressure contours of the solution are given. Final

adapted mesh consists of 20790 elements and 10512 nodes. It is seen that, shock

waves and expansion fans are sharply resolved by the adapted grid and as a result

discontinuities are captured with a high level of accuracy.

Figure 3-38: Adapted mesh and pressure contours of the scramjet inlet problem

Pressure coefficient along the wall, compared with the results of Hosseini et al. [37]

is given in Figure 3-39. Adaptive LSFEM results are in good agreement with the

literature data. Discontinuities are sharply captured without oscillatory behavior by

LSFEM with AMR algorithm.

Convergence history of the adapted solution is given in Figure 3-40. The jump at

iteration 300 corresponds to the change in the time step and others occurred due to

mesh refinement.

P (kPa): 25 30 35 40 45 50 55 60 65 70 75� (kPa):

57

Figure 3-39: Pressure coefficient along the wall of the scramjet inlet

Figure 3-40: Convergence history of the adapted solution for the scramjet inlet problem

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

cp

x

adapted solution

Hosseini et al. [37]

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0 50 100 150 200 250 300 350 400 450 500

log(R)

Number of iterations

58

CHAPTER 4

CONCLUSION & FUTURE WORK

4.1 Conclusion

In this thesis, least-squares finite element method (LSFEM) is employed to

numerically simulate 2-D planar and axisymmetric flows governed by the

compressible Euler equations. There are many advantages of LSFEM over traditional

FEMs. The governing equations of fluid dynamics are non-self-adjoint equations. In

LSFEM, discretization of these equations yield to symmetric positive definite

matrices which can be solved efficiently by iterative matrix solvers using less

memory and CPU load compared to classical Galerkin FEMs. Another advantage is

the robustness of the least-squares formulation; it works for all flow regimes from

subsonic to supersonic with a unified formulation without need of special treatments

and free parameters to tune.

In order to avoid the use of artificial viscosity for wiggle free shock capturing, h-type

adaptive mesh refinement (AMR) is employed. The developed code can work with

both quadrilateral and triangular elements of linear and quadratic order. The effect of

element type and order on the solution accuracy is inspected in detail. The results

indicate that, both element types resolve the flow features correctly and also

resolution of the discontinuities increased by using quadratic interpolating functions

as expected. Further increase in shock resolution is achieved by employing AMR. It

is seen that very satisfactory results can be obtained by starting with a relatively

59

coarse mesh and applying AMR. Upon inspecting the solution of flow over a bump

problem, underestimation of the Mach number is seen after the trailing edge of the

bump which is due to numerical diffusion. This effect is more apparently seen for the

quadrilateral elements and it is lowest for the adaptive solution.

The effect of time step on numerical diffusion is studied with the supersonic ramp

problem. It is observed that the use of smaller time step decreased the numerical

diffusion and as a result shock waves become thinner, but also oscillatory behavior is

seen in the shock regions. By the use of AMR with a very coarse initial mesh, good

shock resolution without any oscillations can be obtained. It is showed that AMR

refines only the elements at the shock locations and undesired mesh propagation is

avoided which also introduced skew elements in the domain. But it is also seen that

solution is not affected by these skew elements, which indicates the robustness of the

LSFEM with AMR.

4.2 Future Work

In the current study, only mesh refinement was done. But it is planned to implement

a mesh coarsening algorithm for performance considerations. It may be more

efficient to start with a uniform medium density mesh and perform mesh refinement

and coarsening at the same time.

Another fact concerning AMR is that, the h-type refinement is effective especially

for the discontinuities in the flow field. But in the smooth flow regions, using p-type

mesh refinement may give better results compared to h-type refinement. Therefore,

increasing element number and order for the smooth flow regions should be

investigated thoroughly in order to determine the need of combining h-type and p-

type refinement.

60

The developed code is capable of solving 2-D planar and axisymmetric problems,

and it be extended to 3-D problems. The difficulties of achieving this goal seemed to

be related mostly to the modification of AMR.

Another useful study, especially for 3-D problems, will be the parallelization of the

element-by-element Jacobi preconditioned Conjugate Gradient solver. This solution

technique is known to have a very high parallelization potential and there are many

studies in literature that can be consulted [39, 40].

61

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